Eta-products, Eichler integrals, and the level-8 Apery limit

Abstract

We give an independent eta-product derivation of the level-8 Apery limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence satisfying the same cubic recurrence with initial values B_0^{(8)}=0, B_1^{(8)}=1. This value was identified numerically by Almkvist-van Straten-Zudilin and was proved by Golyshev via Beukers's Atkin-Lehner modular method; it was later recomputed by Golyshev-Kerr-Sasaki in the motivic/normal-function framework. The continued fraction PCF((2n+1)(3n^2+3n+1),-n^6) = 8/(7 zeta(3)) already appears in Batut-Olivier and was later rediscovered by the Ramanujan Machine as conjecture Z1. The contribution of the present paper is an explicit rederivation, in the eta-product normalization, of the already-known level-8 Apery limit. We spell out the eta-product verification of the Wronskian identity, the normalization of the Eichler integral, the residue computation of the Fricke period polynomial, and the elementary continuant conversion.